-Depends on how you get the new TN. If you go to US-Canada border to get the new TN only, your old TN will be void at the time when you reenter US using the new TN, but you can ask the old employer and the new employer to issue your TN paperwork, respectively, and then you get both TN I-94 at the US-Canada border, then can have two TN at the same time, thus you still can work for the old TN for a couple of week, and then go to work for the new employer.
Another choice is that your new employer files I-129 to USCIS to get the new TN for you within US, thus it will not affect your old TN validity, but you cannot work for the new employer until the I-129 is approved for the new TN (your employer can use premium processing to get it approved in 15 days).